However, a challenge with using these lowest-order elements is that typical discretisations of the transport equation do not have a satisfactory order of accuracy with respect to the grid spacing, (as discussed by [1]} this should be at least approaching second-order).
One route to circumvent this is to use higher-order finite difference or finite volume methods to build up transport stencils, which is the approach used by [2]}.
Unfortunately finite difference or finite volume methods are not supported in many finite element software systems, where these methods may then be unfeasible.
In any case, as argued by [3]}, finite element discretisations may offer particular advantages for discretising the vector transport equation.
The motivation is then to find finite element discretisations that do deliver improved accuracy, while in this work the computational cost of such schemes are of secondary concern.
